JP2006092421A - Molecular orbital calculation device by elongation method, method thereof, program thereof, and recording medium - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a molecular orbital calculation device by the elongation method which obtains the electron state of a molecule with the use of the self-consistent field method for the elongation method, a method thereof, a program thereof, and a recording medium therefor. <P>SOLUTION: When a Fock matrix of the calculation by the self-consistent field method is created by the molecular orbital calculation device by the elongation method, a two-electron integral (rs¾tu) in an active AO area is approximated to 0 in the range of r', s', t', u'≤0, and in the range of r'>0 and s', t', u'≤0, and is calculated in the range of r', s'>0 and t', u'≤0, in the range of t'>0 and r', u'≤0, and in the range of r', s', t'>0 and u'≤0, where r'=r-n<SB>A</SB>, s'=s-n<SB>A</SB>, t'=t-n<SB>A</SB>, and u'=u-n<SB>A</SB>. A coulomb term and an exchange term are calculated for the two-electron integral between each fragment of the active AO area and a fragment to be added. A coulomb term is calculated for the two-electron integral between a frozen AO area and a fragment to be added. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to a molecular orbital calculation device for an elongation method for obtaining an electronic state of a molecule by an ab initio molecular orbital method, and in particular, an elongation method for obtaining an electronic state of a molecule by using a self-consistent field method for the elongation method. The molecular orbital computing device, the molecular orbital computing method of the elongation method, the molecular orbital computing program of the elongation method, and the recording medium on which the molecular orbital computing program of the elongation method is recorded.

  The physical properties of molecules are closely related to the types of atoms and electronic states that make up molecules. If the electronic state of a molecule is known, for example, a differential structure based on energy coordinates is analytically obtained (so-called energy gradient method), thereby performing a stable structure in which the molecular energy is minimized, a transition state structure, a normal vibration analysis, and the like. be able to. Further, for example, by calculating the potential energy with respect to the reaction coordinate in the reaction of the molecule, the reaction system, the generation system, the reaction intermediate, and the transition state are obtained as the equilibrium point. In addition, vibration spectrum, electron spectrum, dipole moment, ionization potential, polarizability, spin density, and the like can be obtained. Thus, if the electronic state of a molecule | numerator is known, the physical property of various molecules can be calculated | required.

  There is a molecular orbital method as an approximate method for obtaining the electronic state of this molecule based on quantum mechanics. The electronic state of a molecule is represented by a molecular orbital, and this molecular orbital is obtained by solving an equation called a Hartree-Fock-Lautern equation (hereinafter abbreviated as “HFR equation”). This HFR equation is an equation for finding a set of spin orbits that is the best approximation to the ground state when the wave function of the system is approximated by one slater determinant.

  In the molecular orbital method, depending on the degree of approximation used when solving the HFR equation, the empirical molecular orbital method represented by the Hückel method and the extended Huckel method are ignored, and specific terms of two-electron integrals are ignored as sufficiently small values. And semi-empirical molecular orbital methods that use measured values as calculation parameters, and ab initio molecular orbital methods that are all calculated from the first principle without using measured values except for physical constants. . The empirical molecular orbital method and the semiempirical molecular orbital method have the disadvantage that the calculation results depend on the approximation method and parameters, but are not reliable. The non-empirical molecular orbital method does not have such a disadvantage. Is excellent. Application programs for executing such ab initio molecular orbital methods include, for example, Gaussian 94/98/03 (US, Gaussian Inc.), GAMESS (US, NRCC), and the like.

  When actually calculating the electronic state of a molecule using the ab initio molecular orbital method, the calculation time increases dramatically as the number N of atoms constituting the molecule increases. This calculation time is generally said to be proportional to the third to fourth power of the number N of atoms. Therefore, for molecules composed of several atoms, the electronic state of the molecule can be calculated within a practical time, but for macromolecules composed of many atoms such as macromolecules, the practical time It was difficult to calculate the electronic state of the molecule because the electronic state of the molecule could not be calculated.

  Accordingly, some inventors of the present invention have developed an elongation method that calculates the electronic state of an aperiodic polymer. In this elongation method, a monomer as an adduct (fragment) is sequentially added to an oligomer as a starting material (starting cluster) so as to follow the polymerization reaction of the polymer, and the target polymer is extended. Each time a fragment is added, instead of a canonical molecular orbitals (hereinafter abbreviated as “CMOs”) basis, a localized molecular orbital (hereinafter abbreviated as “LMOS”) described later is used. ) To calculate the electronic state of the target polymer sequentially. The molecular orbital method for obtaining the electronic state of a molecule using this elongation method is disclosed, for example, in Non-Patent Document 1, Non-Patent Document 2, and Patent Document 1, and this will be outlined below.

  FIG. 7 is a flowchart showing a molecular orbital calculation method for obtaining the electronic state of a molecule by the elongation method in the background art. FIG. 8 is a diagram schematically illustrating molecular orbitals in each process in order to explain a molecular orbital calculation method for obtaining an electronic state of a molecule by the elongation method. FIG. 9 is a diagram for explaining calculation when a fragment is added to the active LMO. FIG. 10 is a diagram for explaining the sequential calculation of the elongation method.

  7 to 10, in the molecular orbital calculation method for obtaining the electronic state of a molecule by the elongation method, first, a starting cluster for a target polymer for which the electronic state is to be calculated is determined, and an atomic orbital (Atomic Orbitals in this starting cluster) is determined. Hereinafter, abbreviated as “AOs”.) CMOs based on the basis are obtained (S101). This initial starting cluster is a portion of the target polymer consisting of the number of atoms from one end of the target polymer to the length at which LMOs can be constructed and the electronic state can be calculated by a known molecular orbital calculation method. It is. The fact that LMOs can be constructed means that the interaction by an atom at one end of the starting cluster does not substantially reach the atom at the other end, and the interaction by an atom at the other end substantially extends to an atom at its one end. It is not possible. The length of the starting cluster varies depending on the types of atoms constituting the starting cluster, but is usually, for example, 10 angstroms to 20 angstroms.

  Next, AOs bases in the molecular orbitals (Molecular Orbitals, hereinafter abbreviated as “MOs”) of the starting cluster are converted into hybrid atomic orbitals (hereinafter abbreviated as “HAOs”) bases (S102). . The CMOs based on the AOs basis obtained in the process S101 are spread over the entire starting cluster as schematically shown in FIG. 8A, but this conversion causes the starting as shown schematically in FIG. 8B. It exists between each atom of the cluster. This conversion can be calculated by using Expressions 21 to 25.

Here, S _ax is an overlap integral between atom a and atom x (x = b, c, d, e), and S ⁺ _ax is a transposed matrix of S _ax . Hereinafter, similarly, the superscript + indicates that it is a transposed matrix. Superscript bar _U al (l = 1,2,3,4) _is located eigenfunction of S ^ax _{S + ax,} and, lambda ² _al is its eigenvalues. b, c, d, and e are the orbits of the sp ³ hybrid orbitals.

Here, Ux is a transformation matrix, and x = a, b, c, d,.

Here, C is a molecular orbital coefficient based on the original atomic orbital, and C ′ is a molecular orbital coefficient based on the hybrid orbital.

Here, ψ ′ _j is the j-th molecular orbital based on the hybrid orbital basis, and χ _s is the atomic orbital.

Here, F ′ is represented by Expression 25-1, S ′ is represented by Expression 25-2, and C ′ is represented by Expression 25-3.

Next, LMOs based on the AOs base of the starting cluster localized so that the phase of the trajectory becomes large at a specific location are created from the CMOs based on the HAOs base in the starting cluster (S103). In the creation of LMOs based on this AOs base, as schematically shown in FIG. 8C, the phase of the orbit is increased on one end side (the frozen LMO region A, the Frozen LMO part) of the starting cluster to which no fragment is added. in a frozen LMOs (Frozen LMOs) φ _i localized, the other end of the starting cluster fragment is added (active LMO region B, active LMO part) active localized as orbit phase becomes large LMOs (Active LMOs) φ _j are created. In this way, the starting cluster is divided into a frozen LMO region and an active LMO region. The interaction between the starting cluster and the fragment occurs only on the other end (reaction end) side of the starting cluster to which the fragment is added. This is because it is considered that it occurs only to the extent that it can be substantially ignored on the one end side where it is not added. The localization processing for creating LMOs based on the AOs base can be calculated by using Expressions 26 to 31.

Here, α _ij is represented by Expression 31-1, β _ij is represented by Expression 31-2, γ _ij is represented by Expression 31-3, and θ is represented by Expression 31-4. ω is expressed by Expression 31-5.

  Next, MO is calculated when a fragment is added to the starting cluster (S104). The electronic state of a molecule can be obtained, for example, by solving a Fock matrix (F matrix) by a self-consistent field method (SCF method). In the SCF method, first, a new electron density is obtained by diagonalizing the F matrix using the initial electron density. Then, the next new electron density is obtained by diagonalizing the F matrix using the new electron density as the initial electron density. This is repeated until the electron density obtained by initializing the electron density and the electron density obtained by diagonalizing the F matrix substantially coincide. By such a procedure, the F matrix is solved in the SCF method.

In this SCF method, it is usually necessary to diagonalize the F matrix F _AO of the AO basis shown in FIG. 9C, but by using the LMOs basis, the F matrix F _AO is converted into the F matrix F _AO of FIG. since fragments as schematically shown in (B) (Attacking Molecule) is only interact with the active LMOs, inverted L-shaped _{F LMO11, F LMO12, F LMO13} , F LMO21 and _{F LMO31} shown in FIG. 9 (C) Since each element in the area portion can be treated as 0, the SCF method may be executed only in the lower right area portion of F _LMO22 , F _LMO23 , F _LMO32, and F _LMO33 . Therefore, compared with a molecular orbital calculation method that properly handles the entire system, the calculation amount is reduced, and the calculation can be performed efficiently and at high speed.

Here, F _LMO11 is a matrix element related to an interaction in which each orbit of frozen LMOs interacts with its own orbit, and F _LMO12 relates to an interaction in which each orbit of frozen LMOs interacts with each orbit of active LMOs. Each matrix element, _FLMO13, is each matrix element relating to the interaction in which each trajectory of frozen LMOs interacts with each trajectory of the fragment. F _LMO21 is a matrix element related to an interaction in which each orbit of active LMOs interacts with each orbit of frozen LMOs, and F _LMO22 is a matrix element related to an interaction in which each orbit of active LMOs interacts with its own orbit. _Yes , _{FLMO 23} is each matrix element related to the interaction in which each trajectory of the active LMOs interacts with each trajectory of the fragment. F _{LMO 31} is a matrix element relating to an interaction in which each orbit of a fragment interacts with each orbit of the frozen LMOs, and F _{LMO 32} is a matrix relating to an interaction in which each orbit of a fragment interacts with each orbit of the active LMOs. The element, _{FLMO 33,} is each matrix element relating to the interaction in which each trajectory of the fragment interacts with its own trajectory.

  When the fragment is added to the starting cluster, the molecular orbital calculation processing after the fragment addition for calculating the MO is expressed by Equations 32 and 33 when considering the case of the above-mentioned polyethylene. Only the part needs to be calculated.

Here, H ^occ _ij (X, Y) is expressed by Expression 33.

Here, φ ^{(occ, X)} _j is the j-th occupied orbit localized in the region X (X is a frozen region or an active region).

  Next, it is determined whether or not a fragment obtained by adding a fragment to the starting cluster is a target polymer (S105). As a result of the determination, if the target polymer is not the target polymer, the starting cluster obtained in the process S104 is added with a fragment as a new starting cluster, and the process returns to the process S102. In the case of a polymer, the calculation of molecular orbitals is terminated.

  By operating in this way, as shown in FIGS. 10A to 10G, fragments are sequentially added to the starting cluster, and the electronic state is sequentially calculated for each addition. In FIG. 10, an oval symbol is a fragment, for example, a monomer when the target substance for calculating the electronic state is a polymer.

Here, in the terminal portion far away from the active LMO region (the frozen AO region), since it does not contribute to the interaction with the fragment, the electronic state can be fixed. Therefore, from the calculation object in the repetitive calculation from processing S102 to processing S105. Can be removed. Accordingly, the repetitive calculation from the processing S102 to the processing S105 is executed in an area of a certain length (active AO area), and this active AO area is added to the other end to which a fragment is added every time a fragment is added. It will move (shift) sequentially. For this reason, the electronic state after the addition of the fragment can be efficiently calculated without degrading the calculation accuracy. In the frozen AO region, the interaction with the fragment in the frozen LMO has a predetermined threshold (for example, 10 ⁻⁵ au, 10 ⁻⁶ au, au means an atomic unit). This is an area that is:

  In the example schematically shown in FIG. 10, the starting cluster consists of two fragments as shown in FIG. 10 (A), and the active AO region is composed of five fragments as shown in FIG. 10 (D). ing. Therefore, as shown in FIG. 10 (E), when there are 6 fragments, a frozen AO region of one fragment is generated, and fragments are added as shown in FIGS. 10 (F) and (G). Each time the frozen AO region is sequentially extended to the other end to which the fragment is added, the active AO region is sequentially shifted to the other end to which the fragment is added.

  Here, in FIG. 10, the oval symbol indicated by hatching indicates a fragment of the active AO area, and the oval symbol indicated by white outline indicates a fragment of the frozen AO area. In the example shown in FIG. 10, the trajectory is localized in the two fragments at the end to which the fragment is added, and the trajectory is localized in the one at the other end to which the fragment is added. LMOs and active LMOs are formed. A region where the frozen LMOs and the active LMOs are formed, that is, a region composed of the frozen LMO region A and the active LMO region B (region of three fragment lengths in FIG. 11) is referred to as a localized region. .

In this way, the molecular orbital calculation method for obtaining the electronic state of a molecule by the elongation method according to the background art assumes a structure in which fragments are sequentially added to a starting cluster for a macromolecule whose electronic state is to be calculated, and MOs on the starting cluster are assumed. With respect to, the LMOs on the starting cluster localized in the active LMO region that interacts strongly with the MOs of the fragment by appropriate unitary transformation are created, and the eigenvalue problem is solved by the SCF method together with the CMOs on the fragment. This is a method of calculating the electronic state of the whole molecule.
Akira Imamura, Yuriko Aoki, and Koji Maekawa “A theoretical synthesis of polymers by using uniform localization of molecular orbirals: Proposal of an elongation method”, J. Chem. Phys., Vol. 95, pp5419-5431 (1991) Yuriko Aoki, “Project Research Contents”, [online], Internet <http://aoki.cube.Kyushu-u.ac.jp/text/contents/JST_project/JST_content_new.html>, [August 31, 2004 Day search] JP 2003-012567 A

  By the way, when the electronic state of a molecule is obtained by the elongation method, when the HFR equation is solved, the SCF method is usually simply used, so there is room for improvement in order to shorten the calculation time.

  The present invention has been made in view of the above circumstances, and is an elongation method molecular orbital calculation device, an elongation method molecular orbital calculation method, and an elongation method molecule that can be analyzed at higher speed than the background art. It is an object of the present invention to provide a recording medium in which an orbital calculation program and an elongation method molecular orbital calculation program are recorded.

In order to achieve the above object, according to one aspect of the present invention, the molecular orbital computing device of the elongation method for obtaining the electronic state of the molecule using the self-consistent field method for the elongation method is the self-consistent field. In creating the Fock matrix in the arithmetic operation, the two-electron integral (rs | tu) of the active AO region is r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , When u ′ = u−n _A (n _A is the number of the end segment of the frozen orbit) is defined, a range satisfying r ′, s ′, t ′, u ′ ≦ 0 and r ′> 0 and s ′, t ′ , U ′ ≦ 0 is approximated as 0, r ′, s ′> 0 and t ′, u ′ ≦ 0, t ′> 0 and r ′, u ′ ≦ 0, and r A range satisfying ', s', t '> 0 and u' ≦ 0 is calculated, and each fragment of the active AO area and the fragment to be added are calculated. The two-electron integral (rs | tu) is calculated with the Coulomb term and the exchange term, and the two-electron integral (rs | tu) between the frozen AO region and the additional fragment is a cutoff with which the Coulomb term is calculated. A process is executed.

And in order to achieve the above-mentioned object, the molecular orbital calculation method of the elongation method according to another aspect of the present invention, which uses the self-consistent field method for the elongation method, When creating the Fock matrix in the operation of the self-consistent field method, the two-electron integral (rs | tu) of the active AO region is r ′ = r−n _A , s ′ = s−n _A , t ′ = t −n _A , u ′ = u−n _A (where n _A is the number of the end segment of the frozen orbit), a range that satisfies r ′, s ′, t ′, u ′ ≦ 0, and r ′> 0 and s A range satisfying ', t', u '≤ 0 is approximated as 0, and a range satisfying r', s '> 0 and t', u '≤ 0, t'> 0 and r ', u' ≤ 0 A range to be satisfied and a range to satisfy r ′, s ′, t ′> 0 and u ′ ≦ 0 are calculated, and each of the fragments in the active AO area is added to the frame. The two-electron integral (rs | tu) is calculated by the Coulomb term and the exchange term, and the two-electron integral (rs | tu) between the frozen AO region and the additional fragment is calculated by the Coulomb term. The cut-off process is executed.

In order to achieve the above object, according to another aspect of the present invention, there is provided an elongation method for causing a computer to obtain an electronic state of a molecule using a self-consistent field method as an elongation method. When the molecular orbital calculation program creates the Fock matrix in the calculation of the self-consistent field method, the two-electron integral (rs | tu) of the active AO region is r ′ = rn− _A , s ′ = s−. When n _A , t ′ = t−n _A , u ′ = u−n _A (n _A is the number of the end segment of the frozen orbit) is defined, a range that satisfies r ′, s ′, t ′, u ′ ≦ 0 And r ′> 0 and a range satisfying s ′, t ′, u ′ ≦ 0 is approximated as 0, a range satisfying r ′, s ′> 0 and t ′, u ′ ≦ 0, t ′> 0 and r A range satisfying ', u' ≦ 0 and a range satisfying r ′, s ′, t ′> 0 and u ′ ≦ 0 are calculated and active A The two-electron integral (rs | tu) between each fragment in the region and the fragment to be added is calculated as the two-electron integral (rs | tu) between the frozen AO region and the fragment to be added. ) Is characterized by executing a cut-off process in which a coulomb term is calculated.

Furthermore, in order to achieve the above-described object, according to another aspect of the present invention, an elongation method for causing a computer to execute a method for obtaining an electronic state of a molecule using a self-consistent field method as an elongation method. When a computer-readable recording medium storing a molecular orbital calculation program creates a Fock matrix in the calculation of the self-consistent field method, the two-electron integral (rs | tu) of the active AO region is r ′ = r If we define −n _A , s ′ = s−n _A , t ′ = t−n _A , u ′ = u−n _A (where n _A is the number of the end segment of the frozen orbit), r ′, s ′, t A range satisfying ', u' ≦ 0 and a range satisfying r ′> 0 and s ′, t ′, u ′ ≦ 0 are approximated as 0, and r ′, s ′> 0 and t ′, u ′ ≦ 0. Range satisfying, t ′> 0 and range satisfying r ′, u ′ ≦ 0, and r ′, s ′, t ′> 0 The range satisfying u ′ ≦ 0 is calculated, and the two-electron integral (rs | tu) between each fragment in the active AO region and the fragment to be added is calculated as the Coulomb term and the exchange term, and is added to the frozen AO region. The two-electron integral (rs | tu) between the two fragments is characterized by performing a cut-off process in which the coulomb term is calculated.

Self-consistent in the molecular orbital calculation device of the elongation method, the molecular orbital calculation method of the elongation method, the molecular orbital calculation program of the elongation method and the molecular orbital calculation program of the elongation method having such a configuration When creating the Fock matrix in the field operation, the two-electron integral (rs | tu) of the active AO region is r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , U ′ = u−n _A (where n _A is the number of the end segment of the frozen orbit), a range that satisfies r ′, s ′, t ′, u ′ ≦ 0, and r ′> 0 and s ′, t A range satisfying ', u' ≦ 0 is approximated as 0, a range satisfying r ′, s ′> 0 and t ′, u ′ ≦ 0, a range satisfying t ′> 0 and r ′, u ′ ≦ 0, and A range satisfying r ′, s ′, t ′> 0 and u ′ ≦ 0 is calculated, and the active AO The two-electron integral (rs | tu) between each fragment of the region and the fragment to be added is calculated as the two-electron integral (rs | tu) between the frozen AO region and the fragment to be added, by calculating the Coulomb term and the exchange term. ) Executes a cut-off process in which a coulomb term is calculated. For this reason, the SCF method can be executed by omitting calculations that do not substantially contribute to the calculation result, so that the SCF method can be executed at a higher speed than the background art.

  The cut-off processing according to the present invention can be applied to a molecular orbital computing device of an elongation method that obtains an electronic state of a molecule by using the SCF method for the elongation method, and the SCF method can be executed at high speed.

This embodiment is an embodiment in which the cut-off process according to the present invention is applied to the molecular orbital calculation process after the addition of the fragment in the process S104 in the background art. Further, the localization process in the process S103 in the background technique is devised. , We are trying to speed up even more. That is, when obtaining the electronic state of a molecule by the ^elongation method, the transformation matrix that converts to the canonical molecular orbital-based region localized molecular orbital is Y _CMO ^RLMO and the matrix that represents the canonical molecular orbital of the region atomic orbital basis. Is the translocation matrix of C _RO ^{CMO +} , the density matrix D ^RO of the atomic atomic orbital basis is U and the transformation matrix for erasing the elements in the off-diagonal block by the Jacobian method, and the localized molecular orbital of the atomic orbital basis is If the matrix representing C _AO ^RLMO and the matrix representing the atomic orbital-based canonical molecular orbital are C _AO ^CMO , the atomic orbital-based canonical molecular orbital is defined as a region using Equation 13 and Equation 14 described later. Localization processing to convert to localized molecular orbitals is executed. Then, when creating the F matrix in the calculation of the SCF method, the two-electron integral (rs | tu) of the active AO region is r ′ = r−n _A , s ′ = s−n _A , t ′ = t−. Defining n _A , u ′ = u−n _A (n _A is the number of the end segment of the frozen orbital), a range satisfying r ′, s ′, t ′, u ′ ≦ 0, and r ′> 0 and s ′ , T ′, u ′ ≦ 0 is approximated as 0, r ′, s ′> 0 and t ′, u ′ ≦ 0, and t ′> 0 and r ′, u ′ ≦ 0. A range and a range satisfying r ′, s ′, t ′> 0 and u ′ ≦ 0 are calculated,
The two-electron integral (rs | tu) between each fragment of the active AO region and the fragment to be added is calculated by calculating the two-electron integral (rs) between the frozen AO region and the fragment to be added by calculating the Coulomb term and the exchange term. | Tu) executes a cutoff process in which the coulomb term is calculated.

Hereinafter, such an embodiment according to the present invention will be described with reference to the drawings. In addition, the structure which attached | subjected the same code | symbol in each figure shows that it is the same structure, The description is abbreviate | omitted.
(Configuration of the embodiment)
FIG. 1 is a block diagram illustrating a configuration of an elongation method molecular orbital calculation apparatus according to an embodiment. FIG. 2 is a diagram for explaining how to obtain the transformation matrix T for converting the density matrix D ^{OAO based} on the orthogonal atomic orbitals into the region atomic orbitals RO.

  In FIG. 1, the molecular orbital computing device 1 of the elongation method includes an arithmetic processing unit 11, an input unit 12, an output unit 13, an internal storage unit 14, an auxiliary storage unit 16, and a bus 18.

  The arithmetic processing unit 11 includes, for example, a microprocessor and its peripheral circuits, and functionally includes a molecular orbital calculation unit 111, a region localization calculation unit 112, and a post-addition molecular orbital calculation unit 113. According to the control program, the input unit 12, the output unit 13, the internal storage unit 14, and the auxiliary storage unit 16 are controlled according to the function.

  The molecular orbital calculation unit 111 calculates CMOs based on the AO base in the starting cluster by a known molecular orbital calculation method. This known molecular orbital calculation method is disclosed in, for example, “Mr. Kaeda,“ Chemical New Series Molecular Orbital Method ”, Hankabo, published on April 30, 1999, first edition”.

  The region localization calculation unit 112 directly converts CMOs based on the AO base in the starting cluster into CMOs based on the LMO base using the conversion matrix Y described later.

  This conversion matrix Y can be obtained as follows. First, the density matrix D of the AO basis is expressed as in Equation 1.

Here, C is a canonical molecular orbital. The superscript indicates the new state, and the subscript indicates the base. The same applies to the following. Therefore, C _AO ^CMO is an AO-based canonical molecular orbital. D is a diagonal occupation number matrix. And the conversion from AO to CMO is defined by Equation 2. Since the number of alphabets is limited, the characters in the background art and the characters in the best mode for carrying out the invention do not necessarily have the same meaning.

Here, φ _i ^CMO is the i-th canonical orbit, and χ _m ^AO is the m-th atomic orbit.

For a restricted Hartree-Fock wave function, the occupancy number is either 2 or 0 depending on whether it is double occupied or unoccupied. Equation 1 satisfies Equation 4 when CMO satisfies Equation 3 which is the standard orthogonal condition together with ^AO overlap integral SAO.

Here, if the non-orthogonal atomic orbital base is converted to the orthogonal atomic orbital base (OAO), the calculation can proceed smoothly. For that purpose, a method of symmetric orthogonalization of Lefdin with the smallest deviation from the original basis function is used. An original basis function is a canonical orbit generally obtained by a nonorthogonal atomic orbital basis. A conversion matrix X for converting the density matrix D ^AO in the non-orthogonal atomic orbital base into the orthogonal atomic orbital base is obtained by diagonalizing S ^AO and is expressed as Expression 5.

Here, V is an eigenvector of S ^AO and e is an eigenvalue of S ^AO . Therefore, the density matrix D ^OAO based on the orthogonal atomic orbital basis is expressed by Equation 6.

Therefore, Expression 7 is derived from Expression 1, Expression 6, and X ⁺ X = XX ⁺ = S ^AO .

As can be seen from Equation 7, the eigenvalue of D ^OAO must eventually be either 2 or 0. When the eigenvalue is 2, it is a double occupied orbit, and when the eigenvalue is 0, it is an unoccupied orbit.

  Next, an occupied track and an unoccupied track for the frozen region and the active region are obtained. This orbit is referred to as “Regional Localized Molecular Orbitals (hereinafter abbreviated as“ RLMO ”)”.

In Figure 2, To obtain the occupied orbitals and unoccupied orbitals for frozen LMO region and the active LMO region, first, as shown in FIG. 2 (A), (B) , frozen If it is the sub-block of ^{D OAO} LMO The region atomic orbital space is created by separating D ^OAO (A) in the region and D ^OAO (B) in the active LMO region and diagonalizing them. In ^{diagonalization,} eigenvectors ^{T B} eigenvectors ^{T A} and ^{D OAO} of ^{D OAO} (A) (B), the region atomic orbital (Regional Atomic Orbitals, below. Abbreviated as "ROs") in a double occupied orbitals Each track is divided into a single occupied track and an empty track (unoccupied track). Therefore, as shown in FIG. 2 (C), T ^A and T ^B are left sub-blocks SuboA in which each matrix element is composed of elements of double occupied trajectory and single occupied trajectory indicated by “occ” in the figure. , SuboB, and right sub-blocks SubvA and SubvB, each matrix element being composed of elements of the unoccupied orbit indicated by “vac” in the figure. That is, when ^{T A} is up to a by m columns matrix of a row b column is an element of dual occupancy trajectory and singlet occupied orbitals, SuboA is made from 1 to m columns, SubvA from m + 1 column Up to row b. However, m <b and n <d. Further, when ^{T B} until n columns a matrix of c row d column and an element of the double occupancy trajectory and singlet occupied orbitals, SuboB is made from 1 to n columns, SubvB from n + 1 column Up to d columns. The single occupied orbit in the frozen LMO region is a hybrid coupled orbit combined with the hybrid coupled orbit of the single occupied orbit in the active region to form a coupled / anti-coupled pair. Instead, the frozen LMO region By apparently shifting from each single occupied orbit in the single occupied orbit in the active LMO region, all ROs can be approximately either a double occupied or unoccupied orbit. Further, in a non-bonded system in which water molecules are bonded by hydrogen bonds, naturally, there are double occupied orbitals. Therefore, as shown in FIG. 2D, the transformation matrix T for transforming the density matrix D ^{OAO based} on the orthogonal atomic orbitals into the region orbit RO has each element from 1 row 1 column to a row m columns, as shown in FIG. Each element from the first row m + 1 column to the a row m + n column is 0, each element from the first row m + n + 1 column to the a row b + n column is each element of SuboA, and the first row b + n + 1 Each element from column a row b + d column is 0, each element from a + 1 row 1 column to a + c row m column is 0, and each element from a + 1 row m + 1 column to a + c row m + n column is SubB Each element from the a + 1 row m + n + 1 column to the a + c row b + n column is 0, and each element from the a + 1 row b + n + 1 column to the a + c row b + d column is a matrix of SuboB.

If an operator for obtaining T from T ^A and T ^B by the above calculation method is represented by “$”, T can be represented by Expression 8.

The density matrix D of RO from Equation 8 becomes Equation 9, and the matrix for converting from the canonical molecular orbital CMO to the region atomic orbital RO is given by T ⁺ X in Equation 10.

Therefore, Expression 11 is obtained from Expression 7 and the unitary condition TT ⁺ = T ⁺ T = 1.

The RO given by this equation 11 is substantially equal to the frozen LMO region and the active LMO region except for the portion where the trajectory in the frozen LMO region extends to the active LMO region and the portion where the trajectory in the active LMO region extends to the frozen LMO region. Is not fully occupied or unoccupied orbital. Therefore, between its occupied orbitals and unoccupied orbitals in D ^RO, performed to substantially localize, for example, the result is such that 10 ^-6 and 10 ^-7 or less, a unitary transformation . This is a Jacobi method similar to that performed in, for example, the following document in order to convert a natural bond orbital (NBO) into a localized molecular orbital.
Literature; AERed and F. Weinhold, J. Chem. Phys. 83, pp 1736 (1985)

In density matrix D ^RO region atomic orbital basis, when the elements in the non-diagonal blocks using the transformation matrix U for erasing the Jacobi method satisfies the expression 12, the region localized molecular orbital (RLMO) basis of the density matrix The non-zero element of D is 2. Therefore, the conversion from the AO base to the RLMO base is given by Expression 14 because the conversion matrix Y is expressed as Expression 13. The RLMO base corresponds to the LMO base but is referred to as an RLMO base in order to show that the derivation method is different from the background art.

  The post-addition molecular orbital calculation unit 113 calculates the MO when a fragment is added to the starting cluster. The calculation of the electronic state is performed by solving the F matrix by the SCF method using the cut-off processing according to the present invention in which the calculation of the two-electron integral (rs | tu) is partially omitted when creating the F matrix. Executed. This cut-off process is executed as follows.

That is, in the two-electron integral (rs | tu) of the active AO region, r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , u ′ = u−n _A (n _A Is the number of the end segment of the frozen orbit, it is approximated to 0 in the range satisfying the equations 15-1 and 15-2, and is calculated in the range satisfying the equations 15-3 to 15-5.
r ′, s ′, t ′, u ′ ≦ 0 (Formula 15-1)
r ′> 0 and s ′, t ′, u ′ ≦ 0 (Formula 15-2)
r ′, s ′> 0 and t ′, u ′ ≦ 0 (Formula 15-3)
t ′> 0 and r ′, u ′ ≦ 0 (Formula 15-4)
r ′, s ′, t ′> 0 and u ′ ≦ 0 (Formula 15-5)

  For the two-electron integral (rs | tu) between each fragment in the active AO region and the fragment to be added, the Coulomb term and the exchange term are calculated, and two electrons between the frozen AO region and the fragment to be added are calculated. For the integral (rs | tu), the Coulomb term is calculated.

Density matrix _{D Total} of AO basis after cutoff of formula 16.
D _Total = D ₁ $ D (Formula 16)
Here, D ₁ is a density matrix of a starting cluster (including a starting cluster newly added as a starting cluster in the repetition calculation), and D _Current is a density of the active AO region to which the fragment is added. ΔD is a contribution to the density matrix in the frozen AO region. In the case of FIG. 10E, D ₁ is Formula 16-1, D is Formula 16-2, and B is Formula 16-1, Formula 16-2, Formula 17-2, Formula 17-. 3 is the fragment of the localization region in _a a 4, _{a 5} and _{a 6.}
D ₁ = D ₁ (A ₁ , A ₂ , A ₃ , B)
≒ D ₁ (A ₁ , A ₂ , A ₃ , B, M)
≈D ₁ (A ₁ , A ₂ , A ₃ ) (Formula 16-1)
D = D _Current (A ₂ , A ₃ , B, M) + δD (A ₁ ) (Formula 16-2)
Further, the total energy E _Total of the AO base after the cutoff is expressed by Expression 17.
W _Total = 0.5 × Tr (E _Total D _1Total ) (Formula 17)
Here, W _Total is Equation 17-1, W ₁ is the total energy of the starting cluster (including the starting cluster newly added as a starting cluster by adding a fragment in the iterative calculation), and W _Current Is the total energy of the active AO region to which the fragment is added, and δW is the contribution to the total energy in the frozen AO region. In the case of FIG. 10E, W ₁ is Expression 17-2 and W is Expression 17-3.
W _Total = W ₁ $ W (Formula 17-1)
W ₁ = W ₁ (A ₁ , A ₂ , A ₃ , B) + δW ₁ (A ₁ −M) (Formula 17-2)
W = W _Current (A ₂ , A ₃ , B, M) + δD [A ₁ − (A ₂ , A ₃ , B, M)]
(Formula 17-3)

  The input unit 12 is a device that inputs various commands such as a calculation start instruction of the molecular orbital computing device 1 and various data such as structure data and initial electron density to the molecular orbital computing device 1. is there. The output unit 13 is a device that outputs the command and data input from the input unit 12, the calculation result of the molecular orbital calculation device 1, and the like, for example, a display device such as a CRT display, LCD, organic EL display, or plasma display, A printing device such as a printer;

  The internal storage unit 14 is a so-called working memory that reads the molecular orbital calculation program and control program executed by the arithmetic processing unit 11 from the auxiliary storage unit 16 and temporarily stores each data during execution of the molecular orbital calculation program. For example, a RAM (Random Access Memory) that is a volatile storage element is provided.

The auxiliary storage unit 16 is a device that stores data and programs such as a nonvolatile storage element such as a ROM and an EEPROM, a hard disk, and the like, and a molecular orbital calculation program or a molecular orbital calculation device 1 that calculates molecular orbitals according to the present invention. Each program (not shown) such as a control program for operating the computer and data (not shown) necessary for executing each program such as the _initial electron density D _initial are stored.

  The arithmetic processing unit 11, the input unit 12, the output unit 13, the internal storage unit 14, and the auxiliary storage unit 16 are connected to a bus 18 so that data can be exchanged with each other.

  If necessary, the molecular orbital computing device 1 may further include an external storage unit 15 and a communication interface unit 17 indicated by broken lines in FIG.

  The external storage unit 15 stores data with a recording medium such as a flexible disk, a CD-ROM (Compact Disc Read Only Memory), a CD-R (Compact Disc Recordable), and a DVD-R (Digital Versatile Disc Recordable). An apparatus that performs reading and / or writing, such as a flexible disk drive, a CD-ROM drive, a CD-R drive, and a DVD-R drive. The communication interface unit 17 is connected to a network, and is a device for transmitting and receiving communication signals to and from other servers and clients via the network.

  If each program is not stored, it may be configured to be installed in the auxiliary storage unit 16 via the external storage unit 15 from a recording medium in which these programs are recorded, and a server ( Each program may be downloaded from a network and communication interface unit 17 from a not-shown. In addition, the data to be input to the molecular orbital calculation device 1 in calculating the molecular orbital may be configured to be input to the molecular orbital calculation device 1 via the external storage unit 15 by a recording medium storing this data. In addition, the molecular orbital calculation device 1 may be configured to be input from the client via the network and communication interface unit 17.

Next, the operation of this embodiment will be described.
(Operation of the embodiment)
First, a coordinate system is determined for a calculation target for obtaining a molecular orbital, and structure data of a starting cluster is created based on the determined coordinate system. Then, structural data is input to the molecular orbital calculation device 1 of the elongation method via the input unit 12, and an instruction to start molecular orbital calculation is input to the molecular orbital calculation device 1 via the input unit 12. The structure data is coordinates based on the determined coordinate system of each atom in the starting cluster.

  FIG. 3 is a flowchart showing the operation of the molecular orbital computing device of the elongation method. FIG. 4 is a flowchart showing molecular orbital calculation processing after fragment addition by executing cut-off processing.

  In FIG. 3, when the structure data of the starting cluster and the instruction to start the calculation are input, the molecular orbital calculation unit 111 of the calculation processing unit 11 calculates CMOs based on the AO base in the starting cluster, and the calculated result is localized. Notification to the processing unit 112 (S11).

  Next, upon receiving this notification, the localization processing unit 112 of the arithmetic processing unit 11 converts the CMOs of the AO base in the starting cluster into the RLMOs base using the above-described transformation matrix Y of Equation 13. Processing is performed and the result is notified to the post-addition molecular orbital calculation unit 113 (S12). As described above, the molecular orbital computing device 1 of the elongation method according to the present embodiment directly converts CMOs of AO bases into RLMOs bases using the conversion matrix Y of Expression 13, and therefore, as in the background art, CMOs There is no need to select two arbitrarily from the above and convert them into MOs localized in the frozen LMO region and the active LMO region, respectively, and repeatedly execute this conversion until convergence. Therefore, the region localization process can be executed at a higher speed than the localization process of the background art. In addition, there is no volatility when CMOs are allocated to MOs localized in the frozen LMO region and the active LMO region, respectively.

Next, when this notification is received, the post-addition molecular orbital calculation unit 113 of the calculation processing unit 11 solves the F matrix using the cut-off processing by the SCF method, thereby calculating the MO when the fragment is added to the starting cluster. Calculate (S13). This processing S13 will be described in detail. In FIG. 4, the post-addition molecular orbital calculation unit 113 first calculates an F matrix using the _initial density matrix D _initial (S21). Next, the post-addition molecular orbital calculation unit 113 calculates the canonical molecular orbital C (B, M) when the fragment is added (S22). Next, the post-addition molecular orbital calculation unit 113 calculates the density matrix D, for example, the density matrix D represented by Expression 16-2 (S23). Next, the post-addition molecular orbital calculation unit 113 determines whether or not the density matrix D has converged (S24). As a result of the determination, if it has not converged (No), the post-addition molecular orbital calculation unit 113 returns the process to the process S21, whereas if it has converged (Yes), the post-addition molecular orbital calculation unit. 113 terminates the molecular orbital calculation process after the addition of the fragment.

Returning to FIG. 3, the post-addition molecular orbital calculation unit 113 determines whether or not the addition of the fragment to the starting cluster is a calculation target (S14). As a result of the determination, if it is not a calculation object, the post-addition molecular orbital calculation unit 113 regards the starting cluster obtained in step S13 as a new starting cluster, and determines the molecular orbital of the new starting cluster. Is notified to the region localization processing unit 112, and the process returns to step S12. On the other hand, the result of the determination, when an arithmetic object is added molecular orbital computing section 113, processing S21 through using the result of the process _S24, calculates the _{D Total} and _{W Total,} calculates the _{E Total,} And these are output to the output part 13, and the calculation of this molecular orbital is complete | finished. Of course, in the processing of processing S12 to processing S14 that is repeated until the fragment obtained by adding the fragment to the starting cluster obtained in this processing S13 becomes an object to be calculated, it does not contribute to the interaction with the fragment as in the background art. By fixing the state, the frozen AO area far away from the active LMO area is excluded from the calculation target, and is executed in the active AO area of a certain length, and this active AO area is fragmented every time a fragment is added. Are sequentially moved (shifted) to the other end side to which is added.

  By introducing the cut-off process in this way, the molecular orbital computing device 1 of the elongation method according to the present embodiment is more than the case of using the background art method more than the case of using the SCF method. Molecular orbitals can be calculated at high speed. For this reason, it is possible to perform calculations within a practical time for macromolecules that could not be calculated within a practical time when using an ordinary simple SCF method or when using a background art method.

  As an example, in the case of calculating the electronic state of polyglycine synthesized from 20 glycines, the comparison results of the calculation times are shown. In this calculation, 5 glycine is used as a starting cluster, 1 to 15 glycines are sequentially added as fragments, and the region is calculated by the molecular orbital calculation method according to the present embodiment without performing the cut-off process. It was executed in the case where only the localization process was executed and in the case where the molecular orbital calculation method according to the background art was used. The molecular orbital computing device was constituted by a personal computer having a Pentium (registered trademark) 4 with a CPU of 3 GHz.

  FIG. 5 is a graph showing a comparison result of calculation times of polyglycine. The horizontal axis in FIG. 5 indicates the number of glycines as fragments, and the vertical axis indicates the SCF calculation time in seconds. A broken line X indicates the SCF calculation time in the case of the present embodiment, and ● indicates each measured value. A broken line Y indicates the SCF calculation time when only the region localization processing is executed without executing the cut-off processing, and ▲ indicates each measured value. A broken line Z indicates the calculation time of SCF in the case of the background art, and ■ indicates each measured value.

  FIG. 6 is a diagram showing the molecular structure of polyglycine. ● represents a hydrogen atom (H), ○ in the diagonally oblique left pattern represents a carbon atom (C), ○ in the diagonally oblique diagonal pattern represents a nitrogen atom (N), and ○ in the lattice pattern represents an oxygen atom ( O).

  As can be seen by comparing the polygonal line Y and the polygonal line Z in FIG. 5, the computation time of the SCF in which only the region localization processing is performed without performing the cut-off processing is shorter than the computation time of the SCF in the case of the background art. It has become. That is, a molecular orbital calculation device that executes only the region localization process without executing the cut-off process is faster than the background art. As the number of fragments increases, the difference between the SCF operation time in the case of the present embodiment and the SCF operation time in the case of the background technology increases, and the background technology increases as the electronic state of the macromolecule is calculated. It is superior to the case of.

  Further, as can be seen by comparing the broken line X with the broken line Z and the broken line Y in FIG. 5, the SCF calculation time in the case of introducing the cut-off process of this embodiment is compared with the SCF calculation time in the case of the background art. Further, it is shorter than the calculation time of the SCF in the case where the above-described Expression 13 is used. That is, the molecular orbital computing device 1 of the elongation method according to the present embodiment in which the cut-off process is introduced is faster than both the background art and the case where the equation 13 is used. As the number of fragments increases, the difference between the SCF calculation time in the case of the present embodiment in which cut-off processing is introduced and the SCF calculation time in the case of the background art is considerably large, and the electronic state of the macromolecule As compared with the background art, the molecular orbital computing device 1 of the elongation method according to the present embodiment is considerably superior.

As another example, poly-glycine glycine as a fragment - were computed by changing the starting cluster energy in the case of synthesizing the _{(CH 3 [CO-NH-} CH 2] N -CO-NH 2). This calculation is performed using five or nine molecular orbital calculation methods that use only STO-3G as a basis function system and perform only the region localization process without executing the molecular orbital calculation method according to the present embodiment and the cutoff process. In each case where the starting glycine was used as a starting cluster, glycine was sequentially added as fragments until the polyglycine was composed based on 20 glycines. The results are shown in Table 1.

Here, the column “Exact” in Table 1 is an energy value obtained by normal molecular orbital calculation for the entire system according to the background art, and the column “Elongation N _st = 5” and “Elongation N _st = 9”. Columns are calculation results with 5 and 9 glycines as starting clusters, and in these columns, molecular orbital calculation methods in which only region localization processing is performed without performing cut-off processing. The calculation result (No cut-off) and the calculation result (cut-off) of the molecular orbital calculation method according to the present embodiment are respectively shown.

  As can be seen from Table 1, as the size (length) of the starting cluster is larger, the calculation result by the molecular orbital calculation method according to the present embodiment substantially matches the accurate energy value. The calculation result by the molecular orbital calculation method according to the present embodiment is equivalent to the calculation result of the molecular orbital calculation method in which only the region localization process is executed without executing the cut-off process.

It is a block diagram which shows the structure of the molecular orbital calculation apparatus of the elongation method in embodiment. It is a figure for ^{demonstrating the calculation} method of the conversion matrix T which converts the density matrix ^DOAO of an orthogonal atomic orbital basis into the area ^| region atomic orbital RO. It is a flowchart which shows operation | movement of the molecular orbital calculating apparatus of an elongation method. It is a flowchart which shows the molecular orbital calculation process after performing a cutoff process and adding a fragment. It is a graph which shows the comparison result of the calculation time of polyglycine. It is a figure which shows the molecular structure of polyglycine. It is a flowchart which shows the molecular orbital calculation method which calculates | requires the electronic state of a molecule | numerator by the elongation method in background art. It is a figure which shows typically the molecular orbital in each process in order to demonstrate the molecular orbital calculation method which calculates | requires the electronic state of a molecule | numerator by the elongation method. It is a figure for demonstrating the calculation in the case of adding a fragment to active LMO. It is a figure for demonstrating the sequential calculation of the elongation method.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Molecular orbital calculation apparatus 11 of an elongation method Operation processing part 12 Input part 13 Output part 14 Internal storage part 15 External storage part 16 Auxiliary storage part 17 Communication interface part 111 Molecular orbital calculation part 112 Area localization calculation part 113 After addition Molecular orbital calculator

Claims (4)

In the molecular orbital computing device of the elongation method, which uses the self-consistent field method for the elongation method to obtain the electronic state of the molecule,
When creating a Fock matrix in the operation of the self-consistent field method,
The two-electron integral (rs | tu) of the active AO region is: r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , u ′ = u−n _A (n _A is frozen) If the end segment number of the orbit is defined, the range satisfying r ′, s ′, t ′, u ′ ≦ 0 and the range satisfying r ′> 0 and s ′, t ′, u ′ ≦ 0 are approximated to 0. A range satisfying r ′, s ′> 0 and t ′, u ′ ≦ 0, a range satisfying t ′> 0 and r ′, u ′ ≦ 0, and r ′, s ′, t ′> 0 and u ′. A range satisfying ≦ 0 is calculated,
The two-electron integral (rs | tu) between each fragment of the active AO region and the fragment to be added is calculated by the Coulomb term and the exchange term,
The molecular orbital computing device of the elongation method, wherein the two-electron integral (rs | tu) between the frozen AO region and the fragment to be added performs a cut-off process in which a coulomb term is computed. In the molecular orbital calculation method of the elongation method, which uses the self-consistent field method for the elongation method to obtain the electronic state of the molecule,
When creating a Fock matrix in the operation of the self-consistent field method,
The two-electron integral (rs | tu) of the active AO region is: r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , u ′ = u−n _A (n _A is frozen) If the end segment number of the orbit is defined, the range satisfying r ′, s ′, t ′, u ′ ≦ 0 and the range satisfying r ′> 0 and s ′, t ′, u ′ ≦ 0 are approximated to 0. A range satisfying r ′, s ′> 0 and t ′, u ′ ≦ 0, a range satisfying t ′> 0 and r ′, u ′ ≦ 0, and r ′, s ′, t ′> 0 and u ′. A range satisfying ≦ 0 is calculated,
The two-electron integral (rs | tu) between each fragment of the active AO region and the fragment to be added is calculated by the Coulomb term and the exchange term,
The molecular orbital calculation method of the elongation method, wherein the two-electron integral (rs | tu) between the frozen AO region and the fragment to be added performs a cut-off process in which a coulomb term is calculated. In the molecular orbital calculation program for the elongation method to be executed by a computer, the electronic state of the molecule is calculated using the self-consistent field method for the elongation method.
When creating a Fock matrix in the operation of the self-consistent field method,
The two-electron integral (rs | tu) of the active AO region is: r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , u ′ = u−n _A (n _A is frozen) If the end segment number of the orbit is defined, the range satisfying r ′, s ′, t ′, u ′ ≦ 0 and the range satisfying r ′> 0 and s ′, t ′, u ′ ≦ 0 are approximated to 0. A range satisfying r ′, s ′> 0 and t ′, u ′ ≦ 0, a range satisfying t ′> 0 and r ′, u ′ ≦ 0, and r ′, s ′, t ′> 0 and u ′. A range satisfying ≦ 0 is calculated,
The two-electron integral (rs | tu) between each fragment of the active AO region and the fragment to be added is calculated by the Coulomb term and the exchange term,
The molecular orbital calculation program of the elongation method, wherein the two-electron integral (rs | tu) between the frozen AO region and the fragment to be added executes a cut-off process in which a coulomb term is calculated. In a computer-readable recording medium in which a molecular orbital calculation program of an elongation method for obtaining a computer is obtained by using a self-consistent field method for the elongation method, and causing the computer to execute the program.
When creating a Fock matrix in the operation of the self-consistent field method,
The two-electron integral (rs | tu) of the active AO region is: r ′ = r−n _A , s ′ = s−n _A , t ′ = t−n _A , u ′ = u−n _A (n _A is frozen) If the end segment number of the orbit is defined, the range satisfying r ′, s ′, t ′, u ′ ≦ 0 and the range satisfying r ′> 0 and s ′, t ′, u ′ ≦ 0 are approximated to 0. A range satisfying r ′, s ′> 0 and t ′, u ′ ≦ 0, a range satisfying t ′> 0 and r ′, u ′ ≦ 0, and r ′, s ′, t ′> 0 and u ′. A range satisfying ≦ 0 is calculated,
The two-electron integral (rs | tu) between each fragment of the active AO region and the fragment to be added is calculated by the Coulomb term and the exchange term,
A two-electron integral (rs | tu) between a frozen AO region and an added fragment executes a cut-off process in which a coulomb term is calculated.

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